﻿ Pauli exclusion principle

# Pauli exclusion principle

The Pauli principle ( also Pauli'sches exclusion principle or Pauli exclusion principle cal ) is a basic principle of quantum mechanics. It was erected in 1925 by Wolfgang Pauli to quantum theoretical explanation of the structure of an atom and said that no electrons can agree on all quantum numbers in an atom. In the modern formulation of the Pauli principle states that the wave function of a quantum system with respect to interchange of identical fermions is anti-symmetric. Since the quarks as the building blocks of protons and neutrons are among the fermions, the Pauli exclusion principle applies to the whole matter in the generally understood sense: fermions " close mutually exclusive ", that can not exist in the same place. The only reason it ever comes to the differentiated structure of matter with atoms, molecules, etc. The Pauli principle therefore determines not only the structure of the atom, but also larger structures; a result, the pressure of the condensed matter of further compression opposes.

## Simplified representation

Starting point of the Pauli principle is the fact that identical particles are indistinguishable in quantum mechanics. For the course of an experiment, or in general the development of a physical system may not change if one is interchanged, two identical particles. Quantum theory, but identical particles arise from interchanging only always the same measurement values ​​when the amount of square (total) wave function remains the same, ie changes at most of the phase component of the wave function. The experimental experience has even shown the further fact that in exchange of two identical particles, the wave function either remains unchanged, depending on the particle type or only changes its sign. Those particles which changes sign, are called fermions. For them, therefore, the wave function is antisymmetric with respect to Teilchenvertauschung. Particles in which the wave function in exchange of the particles remains the same, it is called bosons. The wave function is symmetric with respect to Teilchenvertauschung.

In its special and first observed form stating the Pauli exclusion principle that no two electrons in an atom in all four quantum numbers, which are necessary for its state description in the orbital model match. When two electrons have for example the same major, minor and magnetic quantum numbers, they must therefore be different in the fourth quantum number, the spin quantum number. Since this can only be the values ​​and assume a maximum of two electrons can reside in a single atomic orbital. This fact largely determines the structure of the periodic table.

## General form ( generalized Pauli principle)

### Formulation

The total wave function of a system of identical fermions must be totally antisymmetric with respect to interchange of any two particles P to be:

Here are the place of the spin of the -th fermion and each permutation, which causes the interchange of any two particles, eg for the interchange of the first particle to the second:

### Descriptive interpretation

If we consider a system of two fermions nichtunterscheidbaren so true because of the antisymmetry of the total wave function

For results from the fact, that. Thus, the squared modulus of the wave function, ie the probability density must ensure that one finds in a measurement two fermions in the same place with same spin, be zero.

In many cases ( such a case is, for example, for non-degenerate eigenfunctions of Hamiltonians without spin -orbit coupling is always given ) is the total wave function is represented as a product of spatial wave function and spin wave function, ie

Because of the antisymmetry is then. Is about the spin wave function symmetric, ie, it follows the anti- symmetry of the spatial wave function. Accordingly, it is general that the symmetry of the functions or equivalent to the antisymmetry of the other is. Thus, if the two fermions about the same spin state, then is symmetric and therefore follows the anti- symmetry of the spatial wave function.

These relationships apply, mutatis mutandis, if more than two nichtunterscheidbare fermions are involved.

### Validity

Due to the so-called spin-statistics theorem only fermions with half-integer spin and bosons with integer spin are found in nature. Thus, the Pauli exclusion applies exactly for the particles with half-integer spin.

For bosons, the Pauli exclusion, however, does not apply. These particles satisfy the Bose -Einstein statistics and can have the same quantum states ( in extreme cases up to the Bose -Einstein condensate ).

## Permutational and rotational behavior

The different Permutationsverhalten of fermions and bosons fits for different rotational behavior of the corresponding spinors. In both cases there is a factor of, with the ( ) sign for bosons ( s integer) and the (- ) sign for fermions ( half-integer s ), corresponding to a rotation of 360 °. The connection is obvious, among other reasons, because a permutation of the particles 1 and 2, in fact a complementary rotation of the two particles corresponds to 180 ° (for example, particle 1 to place 2 on the upper semicircle, particle 2 to place 1 on the lower semi-circle).

## Consequences

The Pauli principle leads to the exchange interaction and is responsible for the spin order in atoms ( Hund's rules ) and solids ( magnetism ) responsible.

In astrophysics is explained by the Pauli principle, that old stars with the exception of the so-called black holes - such as white dwarfs or neutron star - not collapse under their own gravity. Here, the fermions produce a back pressure of the pressure deterioration which counteracts a further contraction. This back pressure can be so strong that there is a supernova.

In scattering processes of two identical particles are obtained for the Trajektorienpaar by interchanging always two, although from the outside not distinguishable, but fundamentally different ways, which must be considered in the theoretical calculation of the cross section and scattering wave function.

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